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Krupkova O. — The Geometry of Ordinary Variational Equations
Krupkova O. — The Geometry of Ordinary Variational Equations

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Название: The Geometry of Ordinary Variational Equations

Автор: Krupkova O.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 264

Добавлена в каталог: 14.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\Delta$-prolongation      40
$\Gamma$-prolongation of a vector field      38
0-contact form      35
Action function      42
Adapted chart to a distribution      24
Alternative Lagrangians      123
Anderson — Duchamp — Krupka conditions      61
angular momentum      159
Angular momentum, conservation law      159 165
Anholonomic prolongation of a fibered manifold      30
Annihilator      21
Aristotle group      159
Associated fiber chart      30
Associated fibered coordinates      30 31
Associated system of a closed two-form      25
Autonomous Lagrangean system      71 210
Autonomous semi-finslerian structure      223
Base of a fibered manifold      29
Brachystochrone      3 4
Brachystochrone problem      3
Bunch of final constraint submanifolds      133
Bundle of $\pi_s$-vertical vectors      31
Canonical connection      222
Canonical dynamical two-form      221
Canonical form of a Lagrangean system      67
Canonical form of a Lepagean 2-form      67
Canonical form of Hamilton equations      81 107 108 145
Canonical transformation      187
Cartan connection      224
Cartan distribution      34 37 39
Cartan form      8 47 48 209
Cartan symmetry      172
Cartan theorem      26
Cauchy characteristic      25 191
Cauchy characteristic, vector field      25 191
Characteristic distribution of a closed two-form      25 175
Characteristic distribution of a Lagrangean system      88 107 108 146
Christoffel symbols      121 224
Classical Hamilton — Jacobi equation      18 174 189
Classical Hamilton — Jacobi integration method      18 197
Classification of Lagrangean systems      130
Classification of symmetries      150
Classification of variational semispray connections      227
Codistribution      22
Compatibility conditions      38
Complementary distributions      21
Complete integral      18 174 175
Complete integral of a closed 2-form      176
Complete set of independent first integrals      23 26
Completely integrable connection      38
Completely integrable distribution      23 207
Completely integrable generalized connection      40
Components of a jet field      38
Configuration space      31 63 151
Connection on a fibered manifold      37 38 183 195
Conservation law      149 152 157
Constant of the motion      149 152 160
Constrained system      130
Constraint algorithm      25 130 131 132 134 172 210
Contact form      34
Contact transformation      212
Contactization      34
Contactness      32
Continuous distribution      21
Coordinate free Liouville Theorem      179
Coriolis force      14
Cosymplectic Lagrangean system      99 210
Cotangent representation of symplectic Lagrangean systems      218
Critical section of a Lagrangian      50 51
Curvature of a connection      38
Darboux chart      26 175
Darboux theorem      26 175 176
Decomposition theorem      35
Deformation of a section      42 49
Degenerate Lagrangean system      129
Degree of freedom      12 78
Dirac theory of constrained systems      103 113 134 218 219
distribution      21
Distribution of a constant rank      21 22
Distribution of a locally constant rank      22
Distribution of a non-constrant rank      22
Distribution of class $C^r$      1
Dynamical form      37 44 53 116
Dynamical symmetry      151 171
Eikonal equation      191
Embedding theorem      191
Energy, conservation law      159 165
Energy, one-form      211
Equivalent dynamical forms      117 126
Equivalent Lagrangians      10 61 149
Equivalent systems of a Hamiltonian and momenta      70
Euclidean group      159 162 162 165 168 169
Euler — Lagrange distribution      90 107 108 146 147
Euler — Lagrange distribution, constrained to a submanifold      133
Euler — Lagrange equations      1 12 41 49 51 53 80 209
Euler — Lagrange expressions      3 46
Euler — Lagrange form      41 46 49 209
Euler — Lagrange mapping      41 46 64
Even-order Lagrangean system      100 105
Even-order variational equations      105 183 195
Evolution of a function      100 107 109 214
Exact integration methods for variational equations      148 149 174
Extended dynamics      81 87 129 132 148 217
Extended Lagrangian      85 88 108 109
Extremal of a Lagrangian      50 51 53
Extremal of a locally variational form      53
Fiber chart      29
Fibered manifold      29
Field of extremals      174 190 192
Field of Lagrangean submanifolds      181
Final constraint submanifold      133 218
Finite symmetry      24
Finsler manifold      220 220 223
Finslerian structure      223
First integral      18 23 100 149 152
First integral of a Lagrangean system      152
First integrals in involution      175 177 178
First Theorem of Emmy Noether      157
First variation      2 41
First variation formula      2 41 42 43 47
First variation of the action function      42
First-order Lagrangean system      74 75 101 102 118 122 145 192 214 220
Flat connection      38
Flat generalized connection      40
Foliation      23
Force on a fibered manifold      225
Free energy      12
Free particle on a semi-finslerian manifold      225
Frobenius theorem      24
Froelicher — Nijenhuis bracket      38
Fundamental Lepagean equivalent      49
Galilei group      159 162 166 167 168 169
Galilei transformation      159 163
Gauge function      61 68
General Liouville — Jacobi Theorem      189
Generalized connection      37 40 141 181 181 183
Generalized coordinates      12 78
Generalized dynamical symmetry      172
Generalized Hamilton equations      84 86 88 91 130 145 191 210
Generalized Hamilton — Jacobi equation      180 189
Generalized horizontal form      40
Generalized jet field      40
Generalized Legendre chart      145
Generalized Legendre coordinates      145 214
Generalized Legendre transformation      130 145 148 214
Generalized Liouville Theorem      178 179
Generalized Noether Theorem      160 160 172
Generalized Poincre — Cartan form      49
Generalized Van Hove Theorem      191
Generating function of a canonical transformation      174 187 188
Generating function of acontact transformation      212
Generators of a distribution      21
Geodesic distance in a field of extremals      200
Geodesic distance in a Hamilton — Jacobi field      200
Geodesies      10
Geometric classification of Lagrangean systems      96 129 130
Global extended dynamics      131
Global inverse problem of the calculus of variations      64 65
Global Lagrangian      54 65
Global proper dynamics      131
Globally variational form      53 54
Grifone connection      223
Hamilton equations      17 80 84 107 108 145 210 211 213
Hamilton equations for autonomous Lagrangean systems      211
Hamilton equations for semiregular autonomous Lagrangean systems      212
Hamilton form      84 107 108
Hamilton function      17 67 70 71 210 210
Hamilton vector field      90 191 214
Hamilton — Jacobi distribution      192 193 194 195 196
Hamilton — Jacobi equation      18 187
Hamilton — Jacobi equation for fields of extremals      190
Hamilton — Jacobi field      193 194 195
Hamilton — Jacobi integration method      18 174 182 197
Hamilton's variational principle      12 13 15
Hamilton, extremal      84 129 194 212
Hamiltonian      17 67 70 71 210 210
Hamiltonian formalism      82 217
Hamiltonian system      84 88 100 106
Helmholtz conditions      5 8 57 61 64 118 161
Higher-order semispray problem      96 129 132 134 141 212
Hilbert — Einstein Lagrangian      113
Holonomic constraints      12 77
Holonomic Hamilton extremal      86 87 93 98
Holonomic section      30 34
Homomorphism of fibered manifolds      30
Horizontal distribution      32 37 38
Horizontal form      32
Horizontal form of a connection      38
Horizontality      32
Horizontalization      33
Image of the Euler — Lagrange mapping      64
Independent first integrals      23 175
Infinitesimal first variation formula      47
Infinitesimal symmetry      24
Integral first variation formula      47
Integral manifold      23
Integral manifold of maximal dimension      23
Integral mapping of a distribution      22
Integral mapping of a distribution of maximal dimension      23
Integral section of a connection      38
Integral section of a dynamical form      53 116
Integral section of a jet field      37
Integral section of an ideal      189
Integrating factor      118 126
Integration by quadratures      18 175
Integration methods      16 142 148 188 195 207
Integration problem      18 95 129 132 149 157 174
Invariant transformation of a differential form      26
Invariant transformation of a distribution      24
Inverse Legendre transformation      106
Inverse problem of the calculus of variations      4 9 16 61 65 118 123 126 150 180 207
Inverse problem of the calculus of variations for distributions      123
Isomorphism of a fibered manifold      30 75
Isoperimetric problem      1
Jacobi chart      183 196
Jacobi complete integral      183
Jacobi theorem      180 182 183
Jacobi Theorem for even-order variational equations      183
Jacobi Theorem for odd-order variational equations      184
Jacobi Theorem for regular even-order variational equations      186
Jet field      37
Jet manifold      29
Jet prolongation of a fibered manifold      29
k-contact form      35
Kernel of the Euler-Lagrange mapping      64
kinetic energy      12 122 163
Kinetic energy of a metric      221
Lagrange function      1 41 210
Lagrange space      220 223
Lagrangean foliation      181
Lagrangean submanifold      181 196
Lagrangean system      63 81 83 97 126 149
Lagrangean system of order zero      73 75 83 88 90 100 101 102 109 127
Lagrangean system with dynamical constraints      212
Lagrangean system with no dynamical constraints      153
Lagrangean system with no primary dynamical constraints      131
Lagrangean system with no primary semispray constraints      132
Lagrangean system with no secondary semispray constraints      133
Lagrangean system with primary dynamical constraints      131
Lagrangean system with primary semispray constraints      131
Lagrangean system with secondary semispray constraints      133
Lagrangean system, admitting generalized Legendre transformations      144 214 218
Lagrangian      37 41 210
Lagrangian affine in accelerations      74 102
Lagrangian affine in velocities      73 100 101 109
Lagrangian on $R \Times T^r M$      2\0
Legendre chart      105
Legendre coordinates      17 105 148 213
Legendre transformation      17 97 103 105 144 148 213
Lepage mapping of the first kind      46
Lepage mapping of the second kind      58
Lepagean equivalent of a Lagrangian      46
Lepagean equivalent of a locally variational form      58
Lepagean n-form      41 48
Lepagean one-form      41 46 56
Lepagean two-form      55 55 55 81 83 97 150 171 208 209
Levi-Civita connection      224
Line element induced by a minimal-order Lagrangian      200
Liouville form      211
Liouville integration method      18 207
Liouville theorem      175 178 179 182
Local inverse problem of the calculus of variations      61 98 118
Local Lagrangian      54 55
Local Lagrangian section of a fibered manifold      29
Locally variational form      53 54 81 118 161 220
Lorentz force      14 122 226
Manifold of s-jets of local sections      29
Manifold of s-velocities      31
Maximal integral manifold      23
Maximal vertical distribution      32
mc-parameter family of Hamilton-Jacobi distributions      197
Mechanical system      117 126 225
Mechanical system on a semi-finslerian manifold      225
Method of canonical transformations      207
Metric on $R \Times T M$      220
Minimal-order Lagrangian      68 158
Minkowski metric      208
Minkowski space-time      170
Momentum      17 67 70 144
Momentum, conservation law      159 165
Momentum, map      150
Multiplier      118
Newton equations      54 80 121
Newtonian determinism      99
Noether equation      155 161
Noether symmetry      172
Noether theorem      157 158 172
Noether — Bessel — Hagen equation      155 161
Noetherian symmetry      172
Non-holonomic constraints      78
Nonlinear connection      37
Odd-order differential equations      53
Odd-order Euler — Lagrange equations      68 100 102 103 105 184
Odd-order Lagrangean system      68 103 105 183 186 195 197 198 198 217
Odd-order variational equations      68 100 102 103 105 184
of a vector field      31
One-contact form      35
One-parameter family of sections      49
Order of a Lagrangean system      63 63
Order of contactness      36
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